01 · Start with the quantity, not the confidence level
Start with the quantity, not the confidence level
A confidence interval describes uncertainty in an estimated population quantity. In a warehouse, that quantity might be mean dispatch time, the difference between two picking methods, or the proportion of shipments missing a cutoff. Specify it precisely: mean minutes from release to loading for ordinary weekday shipments at this site. An interval for that mean does not answer how long the next shipment will take, or whether the slowest five percent meet the customer promise.
Choose the target population and sampling unit before collecting data. Eight independent dispatch batches provide eight observations. Eight cartons on one pallet may share the same delay and cannot automatically be treated as eight independent observations. Record inclusion rules, date range, exclusions, and whether the process remained stable. A precise interval around a biased convenience sample is still a poor description of the intended population.
02 · Build and understand the interval
Build and understand the interval
For independent observations from a normal population with unknown standard deviation, the familiar mean interval is x̄ ± t1−α/2,n−1 × s/√n. Here x̄ is the sample mean, s is the sample standard deviation, n is the number of independent observations, α is one minus the confidence level, and t is a quantile of Student's t distribution with n−1 degrees of freedom. The standard error s/√n measures sampling variability of the mean; it is different from the spread s of individual observations.
The estimate determines the center, the critical value determines the confidence requirement, and the standard error determines precision. More observations narrow the interval approximately in proportion to 1/√n. Halving its width usually requires about four times the sample size, assuming similar variability and independence. Raising confidence from 95% to 99% widens the interval. Do not choose the confidence level after seeing whether the interval crosses a preferred target.
03 · Worked warehouse example
Worked warehouse example
Consider fictional independent dispatch times of 8, 9, 10, 11, 12, 10, 9, and 11 minutes. The sum is 80, so x̄ = 10. The squared deviations sum to 12; therefore s² = 12/7 = 1.7143 and s = 1.3093 minutes. With n = 8, the standard error is 1.3093/√8 = 0.4629. The 97.5th percentile of t with seven degrees of freedom is 2.3646.
The margin is 2.3646 × 0.4629 = 1.0946 minutes. The 95% confidence interval is 8.9054 to 11.0946 minutes. These calculations use the small sample t method, not a normal critical value of 1.96. If an operational decision requires demonstrating that the mean is below 11 minutes, this two sided interval still includes values above 11. It supports a mean near ten but does not provide the requested level of precision for that particular claim.
The observations are intentionally simple teaching data. Their modest spread does not establish normality or independence. In an actual project, inspect the raw times and run order, investigate unusual delays, and assess whether different shifts belong to the same population before adopting the calculation.
04 · Interpret coverage correctly
Interpret coverage correctly
A frequentist 95% interval is produced by a procedure that would cover the fixed population parameter in about 95% of repeated samples under its assumptions. After this specific interval is calculated, the parameter is fixed and the interval either contains it or does not. Do not say there is a 95% probability that this fixed mean lies inside these particular bounds. A Bayesian credible interval has a different interpretation and requires a probability model and prior assumptions.
The interval is also not a statement that 95% of dispatch times lie between 8.9054 and 11.0946 minutes. Individual shipments vary more than an estimated average. Under the same normal model, a prediction interval for a future independent observation uses s√(1+1/n), giving much wider bounds. A tolerance interval asks yet another question: what range contains a specified fraction of the population with a specified confidence. Keep these three operational uses separate.
05 · Choose the method for the data
Choose the method for the data
The t interval is exact under independent normal observations. Larger samples can support approximate mean inference for many distributions, but severe skew, heavy tails, clustered delays, and strong autocorrelation need special attention. For paired before and after data, calculate an interval for within pair differences. For independent groups with unequal variances, use a Welch interval rather than pooling variance by default. For a defect proportion, a Wilson or an exact binomial interval can be preferable to an ordinary normal approximation, especially near zero or one.
Bootstrap intervals can help for medians or other statistics, but resample at the independent unit. Resampling individual cartons from a few trucks pretends there are more independent trucks than actually observed. A block or cluster bootstrap must respect the dependence structure. No resampling method repairs unrepresentative coverage of peak periods or changes in the measurement definition.
06 · Strong and weak applications
Strong and weak applications
A strong application samples across the operation's intended conditions, checks the measurement system, states the estimand and interval method, and interprets width against a meaningful operational threshold. It reports the sample size and observed variability along with the bounds. If an interval is too wide to support action, the next step is better measurement, additional independent sampling, or narrowing the population to a legitimate decision scope.
A weak application displays only a mean and two decimal places, calls a narrow interval proof of control, or collects hundreds of readings from a single shift while claiming site wide certainty. Another weak application treats overlapping separate group intervals as a definitive test of the difference. The appropriate analysis estimates the difference directly, accounting for pairing or independence. Lack of statistical evidence about a difference does not establish practical equivalence.
07 · Use the result through DMAIC
Use the result through DMAIC
In Analyze, intervals help distinguish a substantial operational effect from a noisy estimate. Define the business threshold beforehand: perhaps a saving of at least one minute per batch. An interval for the saving entirely above one minute provides a different decision basis from an interval barely above zero. A statistically detectable effect can still be too small to justify a costly change.
Carry the estimate, uncertainty, sampling design, and process conditions into Improve. Verify the proposed intervention on new data rather than reusing the same favorable sample. In Control, monitor both process stability and the relevant customer outcome; an old confidence interval does not remain valid forever when volumes, staff, or equipment change. Archive the calculation with the units, independent sample count, and reasons for exclusions so another analyst can reproduce the conclusion.
08 · Plan a precision focused sample
Plan a precision focused sample
If the desired mean margin is half a minute, a rough planning calculation uses n ≈ (1.96s/E)², where E is the desired margin and s is a credible advance estimate of spread. Using s = 1.3093 from the example gives about 26.35, rounded upward to twenty seven independent observations. This normal approximation is only a starting point; use a t based planning method and allowances for missing observations when the sample is small. A pilot spread estimate itself is uncertain. Sampling more cartons within one batch will not achieve the same precision as sampling more independent batches.
Confidence bounds do not correct measurement bias. If every timestamp starts half a minute late, a larger independent sample can yield a very narrow interval around the wrong mean. Verify the start and stop definitions and measurement calibration before investing in additional sampling.
Connections
Related methods and phase use
Placed under Analyze because of the outcome it supports. Use the method in other phases when its purpose and assumptions fit the work; the phase list is a guide to selection, not a compulsory sequence.
Sources
Professional foundations and further reading
The explanation and fictional teaching examples are written for LeanTrainingLab. Linked primary sources provide professional foundations; example values and local conventions are stated in the guide. The accepted toolset draws on Six Sigma+Lean Toolset, second edition (Lunau, editor; Springer, 2013). No book text or examples are reproduced here.